Group analysis of a one-dimensional kinetic equation and the moment system closure problem. From invariant relations condition to the group stratification method
A. V. Borovskikha,
K. S. Platonovab a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
b Scientific and Educational Mathematical Center, Khetagurov North Ossetian State University, Vladikavkaz, Russia
Abstract:
The problem of establishing a connection between kinetic equations and equations of a continuous medium by the group methods of analysis of differential equations is investigated. In continuation of works [1-7], the simplest, one-dimensional, case is considered. In this paper equations with a submaximal (three-dimensional) symmetry group are studied.
It turns out that the presence of an external force field requires a radical revision of the problem statement not only in terms of using group methods, but also in the general context. Namely, it turns out that it is impossible to consider the equations of a continuous medium as a reduced system of moment equations, since, generally speaking, the high-order moments (in our case, higher than the first) do not exists at all.
However, the use of group methods allows us to overcome this crisis situation, from the point of view of widely accepted ideas, by moving to a practically equivalent formulation of the problem, which does not touch upon either the moment system or the non-existent moment quantities.
The formulation of the problem transformed in this way turned out to be related to the well-known method of group stratification, and on the one side, somewhat expanding the understanding of this method itself, and on the other side, providing a simple and effective general method for solving the problem of the relationship between kinetic equations and the equations of a continuous medium.
Keywords:
kinetic equation, continuous medium equations, invariants, method of group stratification.
PACS:
02.20.-a,
47.45.Ab
MSC: 35Q20,
22E70 Received: 25.07.2025
Revised: 09.08.2025
DOI:
10.4213/tmf11058
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